Daedric summoning was the only way to interact with the Daedric Princes in TES:II, the game which they were also introduced in. From what we know of the Breton these summonings are very in their character, fitting much better than the organized religion/cult style of the Daedric statues in TES:III and TES:IV. This is why I am suggesting that this element remain a part of High Rock's identity and be enabled as gameplay.

This being said, the summoning in TES:II was rather limited. There was only one way to summon; by paying witch covens and mages guild on the EXACT right day on the yearly calendar. This is pretty constraining and doesn't play well with the style of TES:III. This is why expanding the ways summoning can work would be beneficial. Here are the four summoning methods ThomasRuz and I came up with on discord.

1) Pay a witch coven or mages guild on the summoning day of a Daedric Prince to summon them. This would cost ALOT of money but the player would not need the specific items required for the summoning and it will be a 100% success rate. Witch covens may give cheaper prices but will only do it for the Daedra they are associated with.
2) Find a summoning circle/daedric alter and preform a summoning with the right offering items for the Daedra you are trying to summon on their summoning day. This will be effected by the PC's conjuration skill.
3) Find a summoning circle/daedric alter and preform a summoning with the right offering items for the Daedra you are trying to summon after meeting a certain hidden requirement. This may be for example killing 5 innocent people the day of your attempted summon of Melepha or trying while there is a thunderstorm for Sheogorath. There would be characters and books which mention the methods but they would be very hard to find. This would have a higher base success rate than summoning day rituals but it would still be effected by conjuration.
4) Meeting a certain milestone with a NPC who is either a master conjurer or someone who is very devoted to a certain Daedra and asking them to do the summoning. This will be free of every requirement but the specific offerings needed and will have a 100% success rate. Examples of characters who may have this are a Orc priest of Malacath you need to help in Orsinium or a rouge Conjurer who needs to be charmed in a dungeon. They probably wouldn't do this very often for you or if you have a bad relationship with the Daedra.

ThomasRuz mentioned the idea of adding a book to Tamriel Data called “Daedric Summoning: Witch Rites” which would go into detail about what offerings you need and when the dates are. A failed summoning (wrong day, wrong offering, bad skillcheck) will result in the PC being hit with a negative effect. This negative effect will be associated with the Daedra that the player tried to summon; in the case of both the wrong day and the wrong offering it will just be a random effect from one of the Princes. Lastly, there is a 10% chance that Sheogorath would show up instead of the Daedra you were trying to summon in TES:II and I believe this concept should persist in our summoning model.

i know that this would be a departure from Daggerfall lore, but I feel like it might be a good idea to expand the number of days a given daedra can be summoned per year. Waiting an entire year for the correct date, missing the appropriate day or just not getting your one chance because there's a thunderstorm are not very fun gameplay, and could lead to immersion-destroying behavior (like waiting an entire year for the next opportunity).

You could bump it up to six or even twelve times a year, or tie the day to an irregular event (like thunderstorms for Sheogorath). Changes like these could be explained as "recent advances in the art of summoning", advanced numerological or astrological calculations, or legalistic loopholes like using a different calendar or tradition.

I was thinking that the REAL way to do it would be the hidden method or finding a Daedric associated individual to do it. Both methods would allow the player to do it on ANY day and would also be a reward to the player for finding out more about the Daedra by helping its mortal followers or raiding witch covens for secrets. I understand what you are saying though, maybe we can increase the number to 4 evenly spread throughout the year. I think any more would diminish the importance of the whole affair and make the any day methods less special/useful.

Four days might be best. The thing I fear is that, when giving the player the option to summon the daedra at any time, the summoning days will become completely superfluous: when given the option to simply wait for something or do a fun quest to attain it, most people will choose the quest (and they'd be right to).

From the PM conversation between Thomas and Zinitrad over Daedric Summonings

SpoilerShow

Zinitrad - Today at 9:56 AM
yeah, more or less
a sort of holdover of the ancient nedic druidism and shamanism that featured rituals to various aedra and daedra, some sanctioned form that has survived the shift in culture as a specialized role, though one who's mythos and methods have been shaped significantly by elven lore
Zinitrad - Today at 10:01 AM
this 'sanctioned' interaction by a specialized class (as opposed to the wild and outcast covens and druidic societies) is one of the domains of Archei the ritualist, alongside the general performance of ritualism and ceremony, folk magic traditions, burial, and being the psychopomp for the journey through death
ThomasRuz - Today at 10:03 AM
So a priest of Archei would both be knowledgeable about Aedra/Daedra interaction, and more normal ceremonies?
That makes for an interesting dynamic
Zinitrad - Today at 10:04 AM
like I said, its members of Archei's chapel who I imagined to be the dudes in the temples in daggerfall who would let you talk to daedra
but also, the guy who prepares the special ceremonial wax to be burned in candles on whatever holiday, or at least the guy who watches over this preparation
would theoretically be in some way or another acting in the name of Archei
ThomasRuz - Today at 10:06 AM
We have talked about interaction of Daedra before, but more in the sense of the wild/outcast method, which is very costly
Zinitrad - Today at 10:07 AM
wild/outcast is what I imagine to have more direct interaction with them
Archein ritualism is more like glimpses, gleamings, very controlled but limited invocations and exchanges
ThomasRuz - Today at 10:09 AM
Yes, I do think story-wise and gameplay wise that kind of difference between interaction is great
If you as a player need to commune with Daedra, it allows for the player to choose to go down the 'darker' path, depending on how big the urge is to talk to them
Zinitrad - Today at 10:10 AM
one puts you in greater danger and can greatly piss some people off, but offers you a more direct, but also costly route, the other is safe, controlled, sanctioned, but very limited and ambiguous
ThomasRuz - Today at 10:11 AM
I think that strikes a good balance
'We discussed the Daedric princes and daedric summoning. It was proposed that the player would be able to summon lesser daedra for services, advice, bartering and small quests. There might be rules which the player needs to interact through in order to safely work with the daedra. There was also discussion about the Daedric Princes having a hidden disposition tied to their relationship with the player but this might be best done through crossproject.'
^ From our Discord TLDR
Zinitrad - Today at 10:13 AM
I could imagine something like
needing to find something, but having no idea where to look, and so you're instructed to seek out some form of Azuran oracular guidance
well, from there, you could go to the witch coven upon the spring in the woods, where a decrepit old woman tells you to bring her fingers with which she may point the way, and end up getting a damn specific bit of 'go here'
or you could visit the temple, pay some gold to the local ritualist, and commune with him in a dark little room lit by candles with various incantations marked on the floor, until you gain the vague knowledge that its in a place that looks sort of like this, located in this region, good luck